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Register a new userThe $(k,l)$-Euler theorem and the combinatorics of $(k,l)$-sequences
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Isaac Konan
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In 1997, Bousquet-Melou and Eriksson stated a broad generalization of Eulers distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers-Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$ corresponds to Sylvesters bijection. Subsequently, as an open question, a generalization of their bijection was suggested for the case $k,lgeq 2$. In the spirit of Savage and Yees work, we provide and prove in this paper slight variations of the suggested bijection, not only for the case $k,lgeq 2$ but also for the cases $(k,1)$ and $(1,k)$ with $kgeq 4$. Furthermore, we show that our bijections equal the recursive bijections given by Bousquet-Melou and Eriksson in their recursive proof of the $(k,l)$-lecture hall and finally provide the analogous recursive bijection for the $(k,l)$-Euler theorem.
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The packet routing problem plays an essential role in communication networks. It involves how to transfer data from some origins to some destinations within a reasonable amount of time. In the $(ell,k)$-routing problem, each node can send at most $ell$ packets and receive at most $k$ packets. Permutation routing is the particular case $ell=k=1$. In the $r$-central routing problem, all nodes at distance at most $r$ from a fixed node $v$ want to send a packet to $v$. In this article we study the permutation routing, the $r$-central routing and the general $(ell,k)$-routing problems on plane grids, that is square grids, triangular grids and hexagonal grids. We use the emph{store-and-forward} $Delta$-port model, and we consider both full and half-duplex networks. We first survey the existing results in the literature about packet routing, with special emphasis on $(ell,k)$-routing on plane grids. Our main contributions are the following: 1. Tight permutation routing algorithms on full-duplex hexagonal grids, and half duplex triangular and hexagonal grids. 2. Tight $r$-central routing algorithms on triangular and hexagonal grids. 3. Tight $(k,k)$-routing algorithms on square, triangular and hexagonal grids. 4. Good approximation algorithms (in terms of running time) for $(ell,k)$-routing on square, triangular and hexagonal grids, together with new lower bounds on the running time of any algorithm using shortest path routing. These algorithms are all completely distributed, i.e., can be implemented independently at each node. Finally, we also formulate the $(ell,k)$-routing problem as a textsc{Weighted Edge Coloring} problem on bipartite graphs.
The decay bar B -> bar K* (-> bar K pi) l+ l- offers great opportunities to explore the physics at and above the electroweak scale by means of an angular analysis. We investigate the physics potential of the seven CP asymmetries plus the asymmetry in the rate, working at low dilepton mass using QCD factorization at next-to leading order (NLO). The b ->s CP asymmetries are doubly Cabibbo-suppressed lesssim 1 % in the Standard Model and its extensions where the CKM matrix is the only source of CP violation. Three CP asymmetries are T-odd, and can be O(1) in the presence of non-standard CP violation. The T-even asymmetries can reach O(0.1), limited by the small strong phases in the large recoil region. We furthermore point out an easy way to measure CP phases from time-integrated, untagged bar B_d, B_d -> K* (-> K0 pi0) l+ l- and bar B_s,B_s -> phi (-> K+ K-) l+ l- decays. Analyses of these CP asymmetries can rule out, or further support the minimal description of CP violation through the CKM mechanism. Experimental studies are promising for (super) flavor factories and at hadron colliders.
We present results from a search for the flavor-changing neutral current decays $Bto Kell^+ell^-$ and $Bto K^*ell^+ell^-$, where $ell^+ell^-$ is either an $e^+e^-$ or $mu^+mu^-$ pair. The data sample comprises $22.7times 10^6$ $Upsilon(4S)to Bbar B$ decays collected with the BABAR detector at the PEP-II $B$ Factory. We obtain the 90% C.L. upper limits ${mathcal B}(Bto Kell^+ell^-)< 0.50times 10^{-6}$ and ${mathcal B}(Bto K^*ell^+ell^-)<2.9times 10^{-6}$, close to Standard Model predictions for these branching fractions. We have also obtained limits on the lepton-family-violating decays $Bto Ke^{pm}mu^{mp}$ and $Bto K^{*}e^{pm}mu^{mp}$.
This talk covers recent theoretical progress in exclusive semileptonic rare B-decays at low hadronic recoil. The efficient parametric suppression of the 1/mb corrections in this region provides opportunities to probe the Standard Model and beyond at precision level. Notably, angular analysis allows to simultaneously access electroweak flavor physics and hadronic matrix elements, the latter of which constitute the leading source of theoretical uncertainty. Ratios of B ->K* form factors can already be extracted from present data. A comparison with existing theoretical determinations by lattice QCD and light cone sum rules gives a consistent picture over the whole kinematic range. In the future improved analyses will advance our understanding of non-perturbative methods for QCD and of |Delta B|=1 transitions.
It has been argued recently that transverse asymmetries that are expected to be shielded from the presence of the S-wave (Kpi) pairs originating from the decay of a scalar K0* meson, are indeed affected by this pollution due to the impossibility to extract cleanly the normalization for these observables. In this short note we show how using folded distributions, which is nowadays the preferred method to obtain the information from the 4-body decay mode B-> K*(-> Kpi) l+l-, one can easily bypass this problem and extract the clean observables P_{1,2,3} and also P_{4,5,6} in a way completely free from this pollution including all lepton mass corrections. We also show that in case one insists in using uniangular distributions to extract these observables it is possible to reduce this pollution to just lepton mass suppressed terms. On the contrary, the S_i observables, that are by definition normalized by the full differential decay distribution, will indeed suffer from this pollution via their normalization. Finally, we also present a procedure to minimize the error associated to neglecting lepton mass corrections in the distribution defining a massless-improved limit.
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